The CPCTC theorem states that as soon as two triangles room congruent, their equivalent parts space equal. The CPCTC is an abbreviation used for 'corresponding components of congruent triangles are congruent'.
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|1.||What is CPCTC?|
|2.||CPCTC Triangle Congruence|
|4.||FAQs on CPCTC|
The abbreviation CPCTC is because that Corresponding components of Congruent Triangles are Congruent. The CPCTC theorem says that as soon as two triangles are congruent, climate every corresponding component of one triangle is congruent to the other. This means, as soon as two or much more triangles are congruent then their matching sides and angles are additionally congruent or same in measurements. Permit us understand the definition of congruent triangles and corresponding components in detail.
Two triangle are said to be congruent if they have specifically the exact same size and also the very same shape. 2 congruent triangles have three equal sides and also equal angles through respect to each other.
Corresponding sides median the 3 sides in one triangle space in the same position or spot as in the various other triangle. Matching angles average the three angles in one triangle space in the same position or spot together in the other triangle.
In the given figure, △ABC ≅ △LMN. It means that the three pairs the sides and three pairs of angle of △ABC room equal to the 3 pairs of matching sides and also three bag of corresponding angles the △LMN.
In these 2 triangles ABC and LMN, let us recognize the 6 parts: i.e. The three corresponding sides and also the three equivalent angles. Abdominal muscle corresponds to LM, BC coincides to MN, AC synchronizes to LN. ∠A synchronizes to ∠L, ∠B synchronizes to ∠M, ∠C corresponds to ∠N. And if △ABC ≅ △LMN, then together per the CPCTC theorem, the corresponding sides and also angles room equal, i.e. Abdominal muscle = LM, BC = MN, AC = LN, and ∠ A = ∠L, ∠B = ∠M, ∠C = ∠N.
CPCTC Triangle Congruence
CPCTC states that if two triangles room congruent by any type of criterion, then every the corresponding sides and also angles space equal. Here, us are mentioning 5 congruence criteria in triangles.
|SSS||All the 3 corresponding sides space equal||All the equivalent angles are additionally equal|
|AAS||2 matching angles and the non consisted of side room equal||The other matching angles and the various other 2 equivalent sides are also equal|
|SAS||2 matching sides and also the included angle room equal||The other corresponding sides and the various other 2 matching angles are also equal|
|ASA||2 corresponding angles and the included sides room equal||The other equivalent angles and the other 2 equivalent sides are additionally equal|
|RHS / HL||The hypotenuse and also one foot of one triangle room equal to the equivalent hypotenuse and a leg of the other||The other equivalent legs and also the other two equivalent angles room equal|
To prove CPCTC, first, we need to prove the the 2 triangles room congruent v the assist of any type of one of the triangle congruence criteria. Because that example, consider triangles ABC and CDE in which BC = CD and also AC = CD space given.
Follow the points to prove CPCTCBC = CD and also AC = CD (Given)Thus, △ABC ≅ △EDC; by SAS (side-angle-side) criterionNow the 2 triangles room congruent, therefore, using CPCTC, abdominal muscle = DE, ∠ABC = ∠EDC and also ∠BAC = ∠DEC.
Given below are some necessary notes pertained to CPCTC. Have actually a look!Look because that the congruent triangles keeping CPCTC in mind.Before using CPCTC, present that the two triangles are congruent.
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Example 1: Observe the figure given below and also find the length of LM utilizing the CPCTC theorem, if the is provided that △ EFG ≅ △LMN.
Solution: Given that △ EFG ≅ △LMN. So, we can use the ASA congruence dominion to that which claims that if two corresponding angles and the contained side room equal in 2 triangles, then the triangles will be congruent. Here, two angles are provided which space 30 degrees and 102 degrees such that ∠EFG = ∠LMN and also ∠FEG = ∠MLN. So, by applying the CPCTC to organize we deserve to identify the FE and also ML are the equivalent sides of two congruent triangles △ EFG and also △LMN. Therefore, FE = ML. Hence, the size of side LM is 3 units.
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Example 2: Observe the figure given listed below in which PR = RS and QR is perpendicular to PS. Discover y utilizing the CPCTC theorem.
Solution: First let us prove the △PQR ≅ △SQR,PR = RS (given)QR = QR (common side)∠QRP = ∠QRS (as QR is perpendicular come PS)Therefore, △PQR ≅ △SQR (SAS criterion)PQ = QS ( through CPCTC)
Now together PQ = QS